3.0 - Donald Knuth's Up-Arrow Notation (1976)
Background:
In 1976 Donald Knuth Invented Up-Arrow Notation as an easier way to record the so called Hyper-Operators (those past exponentiation), that was more intuitive.
A Knuth-Operator is a infix-binary operator K such that bKp is defined. b is called the base, and p is called the polyponent. b, and p may be any natural numbers, however, with a caveat that b=0 may be undefined in certain formulations, unless 0^0 is set to 1. Normally however b is a positive integer>1 and p is a natural number.
A Knuth-Operator is of the form ^...^ where there are k consecutive carets. The Number of carets is referred to as the Knuth-Operator's Knuth-Degree.
Operators beyond Exponentiation are referred to as hyper-operators, and may be given special names as follows:
^^ - Tetration
^^^ - Pentation
^^^^ - Hexation
^^^^^ - Heptation
^^^^^^ - Octation
b^...^p w/k ^s , where b,p are naturals, and k is a positive integer, may be defined as follows:
1. b^p = b to the power of p
2. b^...^0 = 1
3. b^...^p = b^...(b^...^(p-1))